The affine \(L_p\)  Pólya-Szegö-type principle is closely related to the \(L_p\)  Brunn-Minkowski theory of convex bodies (see e.g.,
[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]} for additional references). The fact that \(L_p\)  spaces have a natural generalization, known as Orlicz spaces, motivated Lutwak, Yang and Zhang [16]}, [17]} to initiate an extension of the \(L_p\)  Brunn-Minkowski theory to an Orlicz-Brunn-Minkowski theory. The definition of a corresponding addition came later, in the work of Gardner, Hug and Weil [18]}. They developed a very general and comprehensive framework for Orlicz-Brunn-Minkowski theory. This theory has expanded rapidly (see e.g., [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}).
